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Question 27765: Wilson Congruence and Primality Agree at the Specified Modulus

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0054-27765 · q-20261015-advance-20261118-r04-wilson-theorem-0015

Original question and givens

Use exact integer n=16>1. Give (n, prime, running_factorial_residues, factorial_residue, minus_one_residue, Wilson_congruence, instance_equivalence, instance_scope).

  • G001 — Use exact integer n=16>1. Give (n, prime, running_factorial_residues, factorial_residue, minus_one_residue, Wilson_congruence, instance_equivalence, instance_scope).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

Step-by-step derivation

S001 · implies

Before

source roles = [wilson-domain-factorial]

After

derived = [[n, prime, running_factorial_residues, factorial_residue, minus_one_residue, Wilson_congruence, instance_equivalence, instance_scope]]

Conditions: none

List the complete operation/mode field order without JavaScript or answer-shape inference;Boolean,string,null and an unclaimed result remain distinct.

Step identifiers and dependencies

Rule: source_ordered_fields_without_JS · Uses: G001

S002 · calculation

Before

Wilson ORIGINAL factorial ORIGINAL running factorial product1 := mul(1, 1)

After

Wilson ORIGINAL factorial ORIGINAL running factorial product1 = 1

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · implies

Before

source roles = [Wilson ORIGINAL factorial, 16, 1, 1, 1]

After

derived = [0, 1, Yes, Yes]

Conditions: none

Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S002

S004 · calculation

Before

Wilson ORIGINAL factorial ORIGINAL running factorial product2 := mul(1, 2)

After

Wilson ORIGINAL factorial ORIGINAL running factorial product2 = 2

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · implies

Before

source roles = [Wilson ORIGINAL factorial, 16, 2, 1, 2]

After

derived = [0, 2, Yes, Yes]

Conditions: none

Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S004

S006 · calculation

Before

Wilson ORIGINAL factorial ORIGINAL running factorial product3 := mul(2, 3)

After

Wilson ORIGINAL factorial ORIGINAL running factorial product3 = 6

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · implies

Before

source roles = [Wilson ORIGINAL factorial, 16, 3, 2, 6]

After

derived = [0, 6, Yes, Yes]

Conditions: none

Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S006

S008 · calculation

Before

Wilson ORIGINAL factorial ORIGINAL running factorial product4 := mul(6, 4)

After

Wilson ORIGINAL factorial ORIGINAL running factorial product4 = 24

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · implies

Before

source roles = [Wilson ORIGINAL factorial, 16, 4, 6, 24]

After

derived = [1, 8, Yes, Yes]

Conditions: none

Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S008

S010 · calculation

Before

Wilson ORIGINAL factorial ORIGINAL running factorial product5 := mul(8, 5)

After

Wilson ORIGINAL factorial ORIGINAL running factorial product5 = 40

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · implies

Before

source roles = [Wilson ORIGINAL factorial, 16, 5, 8, 40]

After

derived = [2, 8, Yes, Yes]

Conditions: none

Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S010

S012 · calculation

Before

Wilson ORIGINAL factorial ORIGINAL running factorial product6 := mul(8, 6)

After

Wilson ORIGINAL factorial ORIGINAL running factorial product6 = 48

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · implies

Before

source roles = [Wilson ORIGINAL factorial, 16, 6, 8, 48]

After

derived = [3, 0, Yes, Yes]

Conditions: none

Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S012

S014 · calculation

Before

Wilson ORIGINAL factorial ORIGINAL running factorial product7 := mul(0, 7)

After

Wilson ORIGINAL factorial ORIGINAL running factorial product7 = 0

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · implies

Before

source roles = [Wilson ORIGINAL factorial, 16, 7, 0, 0]

After

derived = [0, 0, Yes, Yes]

Conditions: none

Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S014

S016 · calculation

Before

Wilson ORIGINAL factorial ORIGINAL running factorial product8 := mul(0, 8)

After

Wilson ORIGINAL factorial ORIGINAL running factorial product8 = 0

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · implies

Before

source roles = [Wilson ORIGINAL factorial, 16, 8, 0, 0]

After

derived = [0, 0, Yes, Yes]

Conditions: none

Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S016

S018 · calculation

Before

Wilson ORIGINAL factorial ORIGINAL running factorial product9 := mul(0, 9)

After

Wilson ORIGINAL factorial ORIGINAL running factorial product9 = 0

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · implies

Before

source roles = [Wilson ORIGINAL factorial, 16, 9, 0, 0]

After

derived = [0, 0, Yes, Yes]

Conditions: none

Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S018

S020 · calculation

Before

Wilson ORIGINAL factorial ORIGINAL running factorial product10 := mul(0, 10)

After

Wilson ORIGINAL factorial ORIGINAL running factorial product10 = 0

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · implies

Before

source roles = [Wilson ORIGINAL factorial, 16, 10, 0, 0]

After

derived = [0, 0, Yes, Yes]

Conditions: none

Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S020

S022 · calculation

Before

Wilson ORIGINAL factorial ORIGINAL running factorial product11 := mul(0, 11)

After

Wilson ORIGINAL factorial ORIGINAL running factorial product11 = 0

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · implies

Before

source roles = [Wilson ORIGINAL factorial, 16, 11, 0, 0]

After

derived = [0, 0, Yes, Yes]

Conditions: none

Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S022

S024 · calculation

Before

Wilson ORIGINAL factorial ORIGINAL running factorial product12 := mul(0, 12)

After

Wilson ORIGINAL factorial ORIGINAL running factorial product12 = 0

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · implies

Before

source roles = [Wilson ORIGINAL factorial, 16, 12, 0, 0]

After

derived = [0, 0, Yes, Yes]

Conditions: none

Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S024

S026 · calculation

Before

Wilson ORIGINAL factorial ORIGINAL running factorial product13 := mul(0, 13)

After

Wilson ORIGINAL factorial ORIGINAL running factorial product13 = 0

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · implies

Before

source roles = [Wilson ORIGINAL factorial, 16, 13, 0, 0]

After

derived = [0, 0, Yes, Yes]

Conditions: none

Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S026

S028 · calculation

Before

Wilson ORIGINAL factorial ORIGINAL running factorial product14 := mul(0, 14)

After

Wilson ORIGINAL factorial ORIGINAL running factorial product14 = 0

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · implies

Before

source roles = [Wilson ORIGINAL factorial, 16, 14, 0, 0]

After

derived = [0, 0, Yes, Yes]

Conditions: none

Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S028

S030 · calculation

Before

Wilson ORIGINAL factorial ORIGINAL running factorial product15 := mul(0, 15)

After

Wilson ORIGINAL factorial ORIGINAL running factorial product15 = 0

Conditions: none

Substitute original roles and established intermediates exactly,not approximations or internally consistent changed operands;extra teaching algebra adds no source-domain gate.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · implies

Before

source roles = [Wilson ORIGINAL factorial, 16, 15, 0, 0]

After

derived = [0, 0, Yes, Yes]

Conditions: none

Retain the prior residue and full product at every k,then its unique Euclidean remainder in0..n-1.The invariant is value congruent to k! mod n,not an assumed last table entry.

Step identifiers and dependencies

Rule: Wilson_EACH_running_factorial_exact_Euclidean_remainder · Uses: G001, S030

S032 · implies

Before

source roles = [16, [[1, 1], [2, 2], [3, 6], [4, 8], [5, 8], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0]]]

After

derived = [No, 0, 15, No, Yes]

Conditions: none

Qualify primality and congruence separately.Both may be False at a composite while equivalence is True.The converse has a separate argument:a proper divisor d>=2 of composite n divides both n and(n-1)!;a congruence to-1 would force d to divide1,a contradiction.This reasoning is not a single instance or finite-cutoff test.

Step identifiers and dependencies

Rule: Wilson_prime_classification_instance_flags_AND_separate_converse_reason · Uses: G001, S031

Reverse checks

  • S032 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S031 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S030 · premise_and_result_audit — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product15 with exact rational arithmetic, not rounded decimals.
  • S029 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S028 · premise_and_result_audit — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product14 with exact rational arithmetic, not rounded decimals.
  • S027 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S026 · premise_and_result_audit — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product13 with exact rational arithmetic, not rounded decimals.
  • S025 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S024 · premise_and_result_audit — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product12 with exact rational arithmetic, not rounded decimals.
  • S023 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S022 · premise_and_result_audit — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product11 with exact rational arithmetic, not rounded decimals.
  • S021 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S020 · premise_and_result_audit — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product10 with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S018 · premise_and_result_audit — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product9 with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S016 · premise_and_result_audit — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product8 with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S014 · premise_and_result_audit — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product7 with exact rational arithmetic, not rounded decimals.
  • S013 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S012 · premise_and_result_audit — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product6 with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S010 · premise_and_result_audit — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product5 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S008 · premise_and_result_audit — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product4 with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S006 · premise_and_result_audit — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product3 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S004 · premise_and_result_audit — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product2 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.
  • S002 · premise_and_result_audit — Recompute Wilson ORIGINAL factorial ORIGINAL running factorial product1 with exact rational arithmetic, not rounded decimals.
  • S001 · premise_and_result_audit — Reverse-check original data,operands,complete scope and premises;finite witnesses,conditional theorem use and formal proof remain distinct.

Result

(16, No, [[1, 1], [2, 2], [3, 6], [4, 8], [5, 8], [6, 0], [7, 0], [8, 0], [9, 0], [10, 0], [11, 0], [12, 0], [13, 0], [14, 0], [15, 0]], 0, 15, No, Yes, Wilson_equivalence_instance)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: f595b68087e9c5908c90d5c5134052b0a472359c001cb28615726a051de02091
  • candidate record SHA-256: 7e5ffd960e173f277c7d35558baa42267a37e9c3620086f339ad9f076a0fb561
  • semantic review receipt SHA-256: 0a76d8e93c08bb16998a0483f3dfc0504f269fc3e1982e6cd868cab0a2856d54
  • question-first witness SHA-256: 09127b736c225714dfa0b7ceb0e6a6e66df27438b480cad8217d3c2f99ce9437
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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